Stability of zonal flows
235
10.6. Exercises on chapter 10
(10.1) Barotropic instability
As we have seen in section 10.2 a necessary, but not sufficient, condition
for barotropic instability of zonal flows is the Kuo criterium. This criterium
implies that there can be no growth of perturbations on a background flow
U = U (y) if U ′′ − β does not change sign with the flow domain. Consider
now the dimensionless zonal flow
U (y)= ¯
U (3y − 2y
3 )
on the domain −1 ≤ y ≤ 1 on a β plane, where ¯
U is a constant amplitude.
a. Show that this flow is stable when | ¯
U | is smaller than β/12.
b. Is this flow unstable on an f -plane?
(10.2) Baroclinic instability
Consider a dimensionless background flow with horizontal velocity field
u = u(z)=a(z +1), v =0, with a>0.
a. Determine the density field that is compatible with this velocity field; use
quasi-geostrophic theory.
b. Calculate the angle between the isopycnals and the horizontal when it is
assumed that the Burger number is constant.
(10.3) Eddy scales
A typical value of the buoyancy frequency N in the Gulf Stream region is
N =10 −3 s −1 .
a. Determine the wavelength of the perturbation with the largest growth factor
in the Eady model for the Gulf Stream.
b. Why do we see (in general) eddies with a larger diameter?
235
10.6. Exercises on chapter 10
(10.1) Barotropic instability
As we have seen in section 10.2 a necessary, but not sufficient, condition
for barotropic instability of zonal flows is the Kuo criterium. This criterium
implies that there can be no growth of perturbations on a background flow
U = U (y) if U ′′ − β does not change sign with the flow domain. Consider
now the dimensionless zonal flow
U (y)= ¯
U (3y − 2y
3 )
on the domain −1 ≤ y ≤ 1 on a β plane, where ¯
U is a constant amplitude.
a. Show that this flow is stable when | ¯
U | is smaller than β/12.
b. Is this flow unstable on an f -plane?
(10.2) Baroclinic instability
Consider a dimensionless background flow with horizontal velocity field
u = u(z)=a(z +1), v =0, with a>0.
a. Determine the density field that is compatible with this velocity field; use
quasi-geostrophic theory.
b. Calculate the angle between the isopycnals and the horizontal when it is
assumed that the Burger number is constant.
(10.3) Eddy scales
A typical value of the buoyancy frequency N in the Gulf Stream region is
N =10 −3 s −1 .
a. Determine the wavelength of the perturbation with the largest growth factor
in the Eady model for the Gulf Stream.
b. Why do we see (in general) eddies with a larger diameter?